If and , then prove that .

Understand the Question We are given the values of and . We need to prove that using the algebraic identity: To use this identity, we first find using . Step 1 · Find the value of Using the algebraic identity Substitute the given values and Step 2 · Evaluate Using the algebraic identity Substitute , , and Hence proved. Answer > Hence proved, . Common Mistakes Sign Errors: Forgetting that when expanding the cubic identity, which leads to adding instead of subtracting . Direct Substitution Error: Trying to substitute directly into without first finding the value of .

Exploring Algebraic Identities | EOT

Question 11

If $a + b + c = 5$ and $ab + bc + ca = 10$ , then prove that $a^3 + b^3 + c^3 - 3abc = -25$ .

Solution

Understand the Question

We are given the values of $a + b + c = 5$ and $ab + bc + ca = 10$ .
We need to prove that $a^3 + b^3 + c^3 - 3abc = -25$ using the algebraic identity: $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
To use this identity, we first find $a^2 + b^2 + c^2$ using $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$ .

Step 1 · Find the value of $a^2 + b^2 + c^2$

Using the algebraic identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$

Substitute the given values $a + b + c = 5$ and $ab + bc + ca = 10$

\begin{aligned} (5)^2 &= a^2 + b^2 + c^2 + 2(10) \\ 25 &= a^2 + b^2 + c^2 + 20 \\ a^2 + b^2 + c^2 &= 25 - 20 \\ a^2 + b^2 + c^2 &= 5 \end{aligned}

Step 2 · Evaluate $a^3 + b^3 + c^3 - 3abc$

Using the algebraic identity $a^3 + b^3 + c^3 - 3abc = (a + b + c)[(a^2 + b^2 + c^2) - (ab + bc + ca)]$

Substitute $a + b + c = 5$ , $a^2 + b^2 + c^2 = 5$ , and $ab + bc + ca = 10$

\begin{aligned} a^3 + b^3 + c^3 - 3abc &= 5 \times (5 - 10) \\ &= 5 \times (-5) \\ &= -25 \end{aligned}

Hence proved.

Answer

Hence proved, $a^3 + b^3 + c^3 - 3abc = -25$ .

Common Mistakes

Sign Errors: Forgetting that $-(ab + bc + ca) = -ab - bc - ca$ when expanding the cubic identity, which leads to adding $10$ instead of subtracting $10$ .
Direct Substitution Error: Trying to substitute directly into $a^3 + b^3 + c^3 - 3abc$ without first finding the value of $a^2 + b^2 + c^2$ .

More questions in EOT

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Q10

If both $x - 2$ and $x - \dfrac{1}{2}$ are factors of $px^2 + 5x + r$ , show that $p = r$ .

Q11

If $a + b + c = 5$ and $ab + bc + ca = 10$ , then prove that $a^3 + b^3 + c^3 - 3abc = -25$ .

Q12

By factoring the expression, check that $n^3 - n$ is always divisible by 6 for all natural numbers $n$ . Give reasons.

Q13

Find the value of

(i) $x^3 + y^3 - 12xy + 64$ , when $x + y = -4$

(ii) $x^3 - 8y^3 - 36xy - 216$ , when $x = 2y + 6$